Dissections, orientations, and trees with applications to optimal mesh encoding and random sampling

We present a bijection between some quadrangular dissections of an hexagon and unrooted binary trees with interesting consequences for enumeration, mesh compression, and graph sampling. Our bijection yields an efficient uniform random sampler for 3-connected planar graphs, which turns out to be determinant for the quadratic complexity of the current best-known uniform random sampler for labelled planar graphs. It also provides an encoding for the set P(n) of n-edge 3-connected planar graphs that matches the entropy bound 1/n log ₂ P(n) = 2 + o(1) bits per edge (bpe). This solves a theoretical problem recently raised in mesh compression as these graphs abstract the combinatorial part of meshes with spherical topology. We also achieve the optimal parametric rate 1/n log₂ P(n, i, j) bpe for graphs of P(n) with i vertices and j faces, matching in particular the optimal rate for triangulations. Our encoding relies on a linear time algorithm to compute an orientation associated with the minimal Schnyder wood of a 3-connected planar map. This algorithm is of independent interest, and it is, for instance, a key ingredient in a recent straight line drawing algorithm for 3-connected planar graphs.